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Data for VSTOXX, VSTOXX futures and various financial and macroeconomic indicators was obtained from Bloomberg. I use two samples: one with 1-minute observation interval and another one with 10-minute interval. The sample period for both samples is from January 11th, 2012 to November 31st, 2012. The subset from January 11th to 31st of August will be used for the in-sample evaluation and the remaining data will be used for out-of-sample evaluation. Public holidays that fall on weekdays were excluded from the dataset. Number of observations in in-sample period of 1-minute data set is 81,539 and in 1-minute set 8241. In full sample the number of observations is 111,742 observations in 1-minute dataset, and in 10-minute data 11,237 observations respectively.

Sep 2012 Aug 2012 Jul 2012 Jun 2012 May 2012 Apr 2012 Mar 2012 Feb 2012 Jan 2012 Dec 2011 Nov 2011 Oct 2011 Sep 2011 0 50000 100000 150000 200000 250000 300000 350000 400000 450000

VSTOXX Mini Futures - Trading Volume and Open Interest

Trading Volume Open Interest

Additionally, I use several explanatory variables to improve modeling, which are summarized in Table 2. Majority of earlier literature has used similar approach for modeling IV. Ahoniemi (2009) tested for several variables, but found only two weekday dummies (for Monday and Friday) and previous returns of underlying S&P500 to be statistically significant. Thus, when speaking of VSTOXX, it is fair to expect the lagged returns of underlying EURO STOXX 50 to be significant. Furthermore, Konstantinidi et al. found that all of the economic variables that they tested for VSTOXX were insignificant. Additionally, altogether only 4 variables were found to be significant when regressed on the 7 volatility indices, which they were studying. DAX and STOXX EUROPE 600 stock indices are used to test for spillover effects between indices, which might also lead to volatility spillover that has been documented across different markets. Additionally, STOXX EUROPE 600 might give information from wider range of stocks. After all, EURO STOXX 50 contains a lot of banks, which has caused the blue chip index to decline much more than the STOXX EUROPE 600 during the financial turmoil of 2012. EUR/USD and EUR/CHF exchange rates and gold price are often used as safe havens during turmoil in other asset classes. Since the VSTOXX is also used as a safe haven for Eurozone stocks, it is justifiable to test if these indicators can help to model implied volatility.

The nature of this study sets certain limitations. Many of the indicators used in previous studies cannot be used, because there is no data available with 1 minute or 10 minute intervals for many indicators. E.g. Euribor interest rates are calculated only once day and thus intraday data is not available. However, these variables have not been statistically significant in majority of related studies6.

6

E.g. Ahoniemi (2009) and Konstatinidi et al. (2007) found 1-month Euribor or U.S interbank interest rates and many other financial and economic variables to be statistically insignificant

Table 2 – Summary of variables tested

The table describes the 8 variables to be tested in the study, as well as whether previous studies have used similar variables for implied volatility modeling.

Variable Explanation Previous

studies DAX A blue chip stock market index consisting of

the 30 major German companies. No STOXX EUROPE 600 A stock market index containing 600

European companies. No

EUR/USD

Euro to US dollar spot exchange rate. Many changes in the world's economy and financial markets reflect to this leading currency rate.

Yes

EUR/CHF

Euro to Swiss franc spot exchange rate. Especially during the financial crisis in Southern Euro countries, Swiss franc has been used widely as a safe haven.

No

Gold

Spot price for Gold forwards from London Metal Exchange. Gold has been used as safe haven for almost all asset classes.

Yes

EURO STOXX 50 EURO STOXX 50 index, for more

information on the index see section 4.1.1. Yes

Weekday dummies

Weekday dummy for all the weekdays. The dummy gets value 1, when is the day indicated by dummy, otherwise zero.

Yes

Hour dummies

Hour dummy for all the trading hours (9:00 CET to 17:00 CET). The dummy gets value 1, when is the hour indicated by dummy, otherwise zero.

No

Appendices 1 and 2 describe all the series and all the data sets. Mean values and standard deviations are very similar in both datasets. During the in-sample-period, VSTOXX reached minimum value of 17.3% and maximum value of 38.3%. The average value was 26.45 % during in-sample period and 25.22 during the combined in-sample and out-of-sample period.

With 1-minute observation intervals, average change in the full period was -0.00009% and the standard deviation of the changes 0.05%. Respectively with 10-minute data, average change was -0.009% and standard deviation of changes 0.19%.

Logged datasets were used to avoid negative volatility. Additionally, Simon (2003) argued that using logs is in-line with positive skewness of IV. As Appendix 2 shows, the data is skewed to right in both data sets. The positive skewness has been reported also in other implied volatility studies. Excess kurtosis is very high in differenced data, as expected. Moreover, Augmented Dickey-Fuller (ADF) tests for unit roots suggest, that first differenced logs are the best choice. The differenced data receives clearly the largest t-values, and the null hypothesis of unit roots can be rejected. For level data the hypothesis could be rejected only on 10% significance level (p-value 0.08). Thus, the peremptory requirement for stationary series is best achieved with the differenced data. Jarque-Bera tests indicate normality of errors in both datasets of VSTOXX in all three forms.

VSTOXX is a very persistent time series and the observations display high auto correlation in both 1-minute and 10-minute series with level and differenced data. Significance is measured with widely-used rule of thumb:

| | _√ , then ρ is statistically significant, where N = number of observations Autocorrelations (AC) are presented in Figures 4 and 5 on the next page. Level data display over 95% autocorrelation for all 36 lags. Differenced log data with 10 minutes intervals has significant autocorrelation for first two lags. The more frequent 1-minute data sets displays autocorrelation for several lags. Additionally, the logged first difference datasets display significant partial autocorrelation (PAC). Partial autocorrelations are represented in Figure 6 on the next page as well. In short, significant PACs suggest AR models to suit data and ACs to MA models respectively. Thus, ACs and PACs indicate that the data should fit very well for the ARMA modeling.

-0.04 -0.02 0 0.02 0.04 1 6 11 16 21 26 31 36

Log VSTOXX first differences - 1min -0.05 0.15 0.35 0.55 0.75 0.95

Log VSTOXX - 1min

-0.1 -0.05 0 0.05 0.1 1 6 11 16 21 26 31 36

Log VSTOXX first differences - 10 min -0.1 0.1 0.3 0.5 0.7 0.9 1 6 11 16 21 26 31 36

Log VSTOXX first differences - 10 min -0.04 -0.02 0 0.02 0.04 1 6 11 16 21 26 31 36

Log VSTOXX first differences - 1min -0.1 -0.05 0 0.05 0.1 1 6 11 16 21 26 31 36

Log VSTOXX first differences - 10 min

Figure 5 – Autocorrelations VSTOXX with 1-minute intervals

Autocorrelations for 36 lags of log VSTOXX and log first differenced VSTOXX with 10-minute intervals

Figure 6 – Autocorrelations VSTOXX with 10- minute intervals

Autocorrelations for 36 lags of log VSTOXX and log first differenced VSTOXX with 10-minute intervals

Figure 7 – Partial autocorrelations

Appendix 2 provides also key statistics for VSTOXX futures during the out-of-sample period. The out-of-sample period consists of 30,203 VSTOXX observations. Nevertheless, close, bid and ask quotes for VSTOXX futures are not available for every minute. The futures dataset consists of about 18,000 bid and ask quotes and of only about 6,000 minute close quotes. The idea of obtaining the futures quotes was to carry through trading simulations with the developed models. However, since there is not data available for each minute, the intraday trading simulations cannot be carried through with futures. In order to measure profit generating abilities of models, the trading simulations should be implemented with same intervals which the model is forecasting. Furthermore, there is no sense to implement the simulations with option straddles, because earlier studies (see, e.g. Brooks and Oozer, 2003) have confirmed that they will not provide any economic profits, and because straddle is not nowadays the best strategy to trade implied volatility.